3.1 - Complex numbers
Arrow Notation
A method of notation for that is used to show that x or f(x) is approaching a particular value
Term
A single number or variable
Polynomial function
Consists of either zero or the sum of a number of a nonzero terms
Reciprocal Function
Function defined on the set of nonzero reals, that sends every real number to its reciprocal
Remainder theorem
If a polynomial(x) is divided by (x-k) then the remainder is the value of f(k)
Maximum Value
If the parabola opens down the vertex represents the highest point on the graph
Minimum Value
If the parabola opens up, the vertex represents the lowest point on the graph
Rational Zero Theorem
If the polynomial has integer coefficients, then every rational zero of f(x) has the form p/q where p is a factor of the constant term asub(o) and q is a factor of the leading coefficient asub(n)
Fundamental theorem
If you have a polynomial of a degree greater than 0, then f(x) has at least one complex zero. We can use when arguing if f(x) is a polynomial degree n>0 and "a" is a non-zero real number then f(x) has exactly n linear factors
Rational Function
Is a function that can be written as the quotient of two polynomial functions
Power Function
Is a function with a single term that is the product of a real number
Turning point
Is a point on a graph that changes the direction from increasing to decreasing or decreasing to increasing
Coefficent
Is a variable raised to a fixed number
Complex Number
Is the sum of a real number and an imaginary number. is expressed in standard form when written in a+bi
Leading term
Is the term containing the highest power of the variable- term with the highest degree
Factor theorem
K is a zero of f(x) and only if (x-k) is a factor of f(x)
Horizontal Asymptote
Line along y-axis that helps describe the behavior of a graph as the input gets very large or very small. Horizontal line that the graph approaches as the input increases or decreases without bound
Continuous function
Meaning has no breaks in a graph - continuous flow
Vertex
One important feature of the graph is that it has an extreme point
Vertical asymptote
Straight lines along the x-axis that can be determined using one sided limits/describe the behavior of a graph as the output gets large/ line that the graph approaches but never crosses
Axis of Symmetry
The graph is also symmetric with a vertical line drawn through the vertex
Parabola
The graph of a quadratic function is a v-shaped curve called a parabola
Degree
The highest power of the variable that occurs in the polynomial
Descartes Rule of Signs
The number of positive real zeros is either equal to the number of sign changes of f(x) or is less than the number of sign changes by an even #
Removable discontinuity
When a graph contains a hole; a single point where the graph is not defined, indicated by an open circle
Complex Conjugate
When dividing a- it can be found by changing the sign of the imaginary part of the complex number ( real part is left unchanged)
Complex Plane
a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary components
Imaginary Number
defined as the square root of (-1)
Multiplicity
the number of times a given factor appears in the factored form of the equation of a polynomial
Direct variation
variation where y=kx^n where k is the constant of proportionality or constant of variatio; Variation where Both variables react the same way
Zeros/roots
x-intercept are the points at which the parabola crosses the x-axis . If they exist, the x-intercepts represent the zeros or roots of the quadratic function ( values of x)
Joint Variation
y=kx/z or y=kxz where k is the constant of proportionality or constant of variation; where one variable is directly or inversely related to more than one variable